A regression of the amount of calories in a serving of breakfast cereal vs. the amount of fat gave the following
results: Calories = 97.1053 + 9.6525(Fat). Which of the following is FALSE?
a) It is estimated that for every additional gram of fat in the cereal, the number of calories
increases by about 10.
b) It is estimated that in cereals with no fat, the total amount of calories is about 97.
c) If a cereal has 2 g of fat, then it is estimated that the total number of calories is about 116.
d) The correlation between amount of fat and calories is positive.
e) One cereal has 140 calories and 5 g of fat. Its residual is about 5 cal.
Which of the following statements is/are true?
I. Correlation and regression require explanatory and response variables.
II. Scatterplots require that both variables be quantitative.
III. Every least-squares regression line passes through ( x , y ).
a) I and II only
b) I and III only
c) II and III only
d) I, II, and III
e) None of the above
Scientists rated the activity level of fish at different temperatures (Celsius). A rating of 0 indicates no activity
and a rating of 100 indicates extremely heavy activity. The data they collected are given in the table below.
Fish act.
Water temp.
82
21
65
24
62
29
90
18
51
29
79
22
87
20
Which of the following statements is true?
a) The level of fish activity helps explain the water temperature. At low levels of fish activity, the water is
cooler. As fish move around more, water temperature increases.
b) Increasing the water temperature causes the fish to swim faster.
c) As water temperature decreases, the level of fish activity increases somewhat constantly.
d) The correlation coefficient, 0.91, indicates that there is a fairly strong positive linear relationship
between level of fish activity and temperature.
e) Based on our sample data, we can safely estimate that the level of fish activity would be about 34 at a
temperature of 12 C .
If data set A of (x, y) data has correlation coefficient r = 0.65, and a second data set B has correlation r = –0.65,
then
a) the points in A exhibit a stronger linear association than B.
b) the points in B exhibit a stronger linear association than A.
c) neither A nor B has a stronger linear association.
d) you can’t tell which data set has a stronger linear association without seeing the data or
seeing the scatterplots.
e) a mistake has been made—r cannot be negative.
Suppose we fit a least-squares regression line to a set of data. What is true if a plot of the residuals shows a curved
pattern?
a) A straight line is not a good model for the data.
b) The correlation must be 0.
c) The correlation must be positive.
d) Outliers must be present.
e) The regression line might or might not be a good model for the data, depending on the extent of the curve.
Which of the following statements about the correlation coefficient is true?
a) The correlation coefficient measures the proportion of variability between the two variables.
b) The correlation coefficient will be equal to 1 only if all the data lie on a perfectly horizontal straight line.
c) The correlation coefficient measures the fraction of outliers that appear in a scatterplot.
d) The correlation coefficient has no unit of measurement and must always lie between –1 and 1, inclusive.
e) The correlation coefficient equals the proportion of times two variables lie on a straight line.
Consider the scatterplot below.
According to the scatterplot, which of the following is a plausible value for the correlation coefficient between
weight and MPG?
a) 1.0 . b) 0.9 . c) 0.5 . d) 0.2. e) 0.7.
John’s parents recorded his height at various ages up to 66 months. Below is a record of the results.
Age (months)
Height (inches)
36
35
48
38
54
41
60
43
66
45
Which of the following is the equation of the least-squares regression line of John’s height on age? (NOTE:
You do not need to directly calculate the least-squares regression line to answer this question.)
a) Height = 12 × (Age).
d) Height = 60 – 0.22 × (Age).
b) Height = 0.34 + 22.3 × (Age).
e) Height = 22.3 + 0.34 × (Age).
c)
Height = Age/12.
What is the slope for the LSRM? Interpret the slope in context of this problem. 0.9383; the left hand span increases
approximately 0.9383 units for every 1 unit increase in right hand span.
Write the LSRM using the appropriate variables.
Left Hand Span = 0. 9383(Right Hand Span) + 1.4365
What is the correlation coefficient? Classify the correlation.
Interpret R-sq in context of the problem.
Interpret s in context of the problem.
𝒓 = √𝟎. 𝟗𝟎𝟐 = 𝟎. 𝟗𝟒𝟗𝟕…Strong Correlation
90.2% of the variation in Left Hand Span can be explained by the LSRM.
The standard deviation of the residuals is 0.6386.
Is it possible to accurately determine a person’s body fat percentage from their waist size? Researchers hoping to find a
good estimate randomly selected 20 male subjects, measured their waists and determined percentage of body fat. The data
is given in the following table.
Waist
(in)
32
36
38
33
39
40
41
35
38
38
33
40
36
32
44
33
41
34
34
44
Body Fat
(%)
6
21
15
6
22
31
32
21
25
30
10
20
22
9
38
10
27
12
10
28
Create a linear LSRM to predict % Body Fat from Waist Size.
Body Fat = 2.22(Waist Size) - 62.56 (Found Using STAT-CALC-4)
State the correlation.
r = .887
What fraction of the variation in Body Fat can be explained by the LSRM?
r2 = 78.7%
Predict the % Body Fat of a person with a…
30 inch waist.
4.04%
50 inch waist.
48.4%
Estimate the waist size of a person having 5% Body Fat.
5 = 2.22(Waist Size) – 62.56 →
30 in
At summer camp, one of Carla’s counselors told her that you can determine air temperature from the number of cricket
chirps.
a) What is the explanatory variable, and what is the response variable? EXP: Cricket Chirps RESP: Air Temp
To determine a formula, Carla collected data on temperature and number of chirps per minute on 12 occasions. She
entered the data into her calculator and did 2-Var Stats. Here are some results:
x = 166.8
s x = 31.0
y = 78.83
s y = 9.11
r = 0.765
𝟗.𝟏𝟏
𝟎. 𝟕𝟔𝟓 (𝟑𝟏.𝟎)
𝒎=
= 𝟎. 𝟐𝟐𝟓, 𝒃 = 𝟕𝟖. 𝟖𝟑 − 𝟎. 𝟐𝟐𝟓(𝟏𝟔𝟔. 𝟖) = 𝟒𝟏. 𝟑 Temp = 0.225(Chirps) + 413
b) Use this information to find the equation of the least-squares regression line. Show your work.
c) Predict the temperature if the number of recorded chirps is…
i.
150
75o
ii.
200
86.3o
d) Determine the number of chirps associated with a temperature of
i.
60o
83 chirps
ii.
95o
239 chirps
The following represents length and weight data for 12 perch caught in a lake in Finland
Weight
Length
Weight
(grams) (cm)
(grams) (cm)
Length
5.9
8.8
300.0
28.7
100.0
19.2
300.0
30.1
110.0
22.5
685.0
39.0
120.0
23.5
650.0
41.4
150.0
24.0
820.0
42.5
145.0
25.5
1000.0
46.6
a) Suppose you want to use the length of a perch to predict its weight. Use your calculator to make an
appropriate scatter plot. Describe any apparent association.
Positive non-linear association.
Original
Transformed
Weight vs Length
Weight vs Length
b) How do you expect the weight of animals of the same species to change as their length increases? Make a
transformation of weight without using logarithms that should straighten the plot if your expectation is correct. Plot
the transformed weights against length. Then find the equation of the least-squares line for the transformed data.
Record the equation below. Define any variables you use.
A residual plot of the original data shows a quadratic pattern. Transform by
taking the square root of the Weight data. The plot above indicates a more linear
pattern. A resulting LSRM would be √𝑾𝒆𝒊𝒈𝒉𝒕 = 𝟎. 𝟖𝟎𝟑(𝑳𝒆𝒏𝒈𝒕𝒉) − 𝟔. 𝟒𝟔𝟔
c) How well does the linear model you calculated in (b) fit the transformed data? Justify your answer with graphical
and numerical evidence. Use your model from (b) to predict the weight of a Finnish perch whose length is 35 cm.
Estimate the length of a Finnish perch weighing 500 grams.
The calculator output shows the LSRM from the transformed data. The correlation
is 0.99, indicating a strong correlation.
For a length of 35 cm, 𝑾𝒆𝒊𝒈𝒉𝒕 = (𝟎. 𝟖𝟎𝟑(𝟑𝟓) − 𝟔. 𝟒𝟔𝟔)𝟐 = 𝟒𝟔𝟖 𝒈
The length of a 500 g fish, √𝟓𝟎𝟎 = 𝟎. 𝟖𝟎𝟑𝒙 − 𝟔. 𝟒𝟔𝟔 = 𝟑𝟔 𝒄𝒎